Just
as Johannes Kepler saw the foundations of harmonics in geometry, which
describes the fundamental laws of space, so Rudolf Stössel also begins
with geometry in his work, and finds many ratios of small whole numbers
there, which form the building blocks for musical intervals. Regarding
the illustrations he writes:

“Cicero, as he told us, found an old crumbling gravestone in the
undergrowth, the carving on which indicated that it was the gravestone
of Archimedes.

“Figure 4 reproduces the drawing and shows the two primal geometric
forms, the square and the circle in their closest relationship, and
also the symmetrical form of the isosceles triangle. If this image is
rotated around its axis of symmetry, three bodies emerge, a cone, a
sphere, and a cylinder. Their volumes have the exact ratio 1 : 2 : 3.
These primal forms correspond to a proportion of the three smallest
whole numbers, to the beginning of the number series, and to
mathematics generally. But they also correspond to the tones c c′ g′,
i.e. the intervals of the octave and fifth, the strongest consonances,
also the primal consonances, so to speak, and to the duodecimal.”

The
two middle rows in the illustration mostly speak for themselves-they
show the many forms in which ratios of small whole numbers emerge from
the combination of the primal forms of the square and circle. Regarding
the lower figure, Stössel writes:

“I now wish to combine elementary forms once again, an equilateral
triangle with its inscribed circle (Fig. 11). We place the compass in
the top corner and draw an arc through the lower corners. This gives us
a large sector. Now we construct the inner circle of the sector, which
touches its radii and the arc. I will leave it to the reader to
construct its center and calculate my result. In any case, the ratio of
the area of the smaller circle to the greater one, and then to the
sector, is 3 : 4 : 6.

“Here we already see the fifth 6/4 = 3/2, the fourth 4/3, and the octave 2/1.
Now we multiply the numbers by 2, getting 6 : 8 : 12, and add the small
sector with the dashed arc, then we have four areas with the ratio
numbers 6 8 9 12-the Harmonia Perfecta Maxima.”